A036896 -id:A036896 - cp's OEIS frontend

A049439 Numbers k such that the number of odd divisors of k is an odd divisor of k.

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 128, 144, 225, 256, 288, 441, 450, 512, 576, 625, 882, 900, 1024, 1089, 1152, 1250, 1521, 1764, 1800, 2025, 2048, 2178, 2304, 2500, 2601, 3042, 3249, 3528, 3600, 4050, 4096, 4356, 4608, 4761, 5000, 5202, 5625, 6084

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR concept formation program.

Sequence consists of all numbers of the form A000079(k)*A036896(m). - Matthew Vandermast, Nov 14 2010

			There are 3 odd divisors of 18, namely 1,3 and 9 and 3 itself is an odd divisor of 18.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics.

Contains A000079 and A036896.
Cf. A001227, A033950.
Subsequence of A028982. Includes A120349, A120358, A120359, A120361, A181795. See also A181794.

a049439 n = a049439_list !! (n-1)
a049439_list = filter (\x -> ((length $ oddDivs x) `elem` oddDivs x)) [1..]
   where oddDivs n = [d | d <- [1,3..n], mod n d == 0]
-- Reinhard Zumkeller, Aug 17 2011

ok[n_] := (d = Length @ Select[Divisors[n], OddQ] ;
  IntegerQ[n/d] && OddQ[d]); Select[Range[6100], ok]
(* Jean-François Alcover, Apr 22 2011 *)
odQ[n_]:=Module[{ods=Select[Divisors[n],OddQ]},MemberQ[ods,Length[ ods]]]; Select[Range[7000],odQ] (* Harvey P. Dale, Dec 18 2011 *)
Select[Range[6000], OddQ[(d = DivisorSigma[0, #/2^IntegerExponent[#, 2]])] && Divisible[#, d] &] (* Amiram Eldar, Jun 12 2022 *)

is(n)=my(d=numdiv(n>>valuation(n,2))); d%2 && n%d==0 \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A000079(k)*A016754(m) for appropriate k, m. - Reinhard Zumkeller, Jun 05 2008

Example corrected by Harvey P. Dale, Jul 14 2011

A057265 Even refactorable numbers (i.e., the number of divisors is itself a divisor and it is also even).

Original entry on oeis.org

2, 8, 12, 18, 24, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 228, 232, 240, 248, 252, 276, 288, 296, 328, 344, 348, 360, 372, 376, 384, 396, 424, 444, 448, 450, 468, 472, 480, 488, 492

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk), Aug 21 2000

Invented by the HR mathematical theory formation program.

			18 is refactorable because tau(18) = 6 and 6 divides 18 and 18 is even.

S. Colton, Unpublished PhD Thesis, University of Edinburgh, 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A033950, A036896.

[k:k in [2..500 by 2]| IsIntegral(k/d) and IsEven(d) where d is #Divisors(k)]; // Marius A. Burtea, Jan 13 2020

rfnQ[n_]:=Module[{ds=DivisorSigma[0,n]},Divisible[n,ds] && EvenQ[ds]];Select[Range[2,500,2],rfnQ]  (* Harvey P. Dale, Mar 14 2011 *)

Corrected (erroneous term 36 removed) by Harvey P. Dale, Mar 14 2011

A181795 Numbers k such that the number of odd divisors of k is an odd divisor of k, and the number of even divisors of k is an even divisor of k.

Original entry on oeis.org

4, 16, 36, 144, 256, 576, 900, 1764, 2304, 2500, 3600, 4356, 6084, 7056, 8100, 10000, 10404, 12996, 17424, 19044, 22500, 24336, 26244, 30276, 32400, 34596, 36864, 41616, 49284, 51984, 57600, 60516, 65536, 66564, 76176, 79524, 90000

Offset: 1

Matthew Vandermast, Nov 14 2010

nonn

All members are even squares (A016742). Intersection of A049439 and A181794.

Includes all numbers of the form A001146(m)*A036896(n) for m>1.

			a(3)=36 has 3 odd divisors (1, 3, and 9) and 6 even divisors (2, 4, 6, 12, 18, and 36). 3 and 6 are odd and even respectively, and both are divisors of 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..96 from Harvey P. Dale)

Subsequence of A000290, A016742, A120351.

See also A033950,A181687. For refactorable members of this sequence, see A120349.

ndQ[n_]:=Module[{d=Divisors[n],od,ev},od=Count[d,?OddQ];ev=Count[ d, ?EvenQ]; ev!=0&&OddQ[od]&&EvenQ[ev]&&Divisible[n,od]&&Divisible[ n, ev]]; Select[Range[100000],ndQ] (* Harvey P. Dale, Feb 24 2016 *)

isok(n) = my(nod = numdiv(n>>valuation(n, 2)), noe = if (n%2, 0, numdiv(n/2))); (nod % 2) && nod && !(n % nod) && !(noe % 2) && noe && !(n % noe); \\ Michel Marcus, Jan 14 2014

More terms from Nathaniel Johnston, Nov 17 2010

A036897 Square root of odd refactorable numbers.

Original entry on oeis.org

1, 3, 15, 21, 25, 33, 39, 45, 51, 57, 69, 75, 81, 87, 93, 111, 123, 129, 141, 159, 177, 183, 189, 201, 213, 219, 225, 237, 249, 267, 291, 303, 309, 315, 321, 327, 339, 343, 381, 393, 405, 411, 417, 447, 453, 471, 489, 495, 501, 519, 525, 537, 543, 567, 573

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

nonn

Odd refactorable numbers are always squares.

			15^2 is refactorable because 225 has 9 divisors and 9 divides 225.

Amiram Eldar, Table of n, a(n) for n = 1..1001
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A033950, A036896.

Select[Range[1, 1000, 2], Divisible[#^2, DivisorSigma[0,#^2]] &] (* Amiram Eldar, Jul 02 2019 *)

isrefac(n) = ! (n % numdiv(n));
lista(nn) = {forstep (n = 1, nn, 2, if (isrefac(n), print1(sqrtint(n), ", ")););} \\ Michel Marcus, Aug 31 2013

a(n) = sqrt(A036896(n)). - Amiram Eldar, Jul 02 2019

A208251 Number of refactorable numbers less than or equal to n.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Wesley Ivan Hurt, Jan 12 2013

A number is refactorable if it is divisible by the number of its divisors.

			a(1) = 1 since 1 is the first refactorable number, a(2) = 2 since there are two refactorable numbers less than or equal to 2, a(3) through a(7) = 2 since the next refactorable number is 8.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number.

Cf. A033950, A000005, A141586, A057265, A036896, A036898, A114617.

Partial sums of A336040.

with(numtheory) a:=n->sum((1 + floor(i/tau(i)) - ceil(i/tau(i))), i=1..n);

Accumulate[Table[If[Divisible[n, DivisorSigma[0, n]], 1, 0], {n, 1,100}]] (* Amiram Eldar, Oct 11 2023 *)

a(n) = sum(i=1, n, q = i/numdiv(i); 1+ floor(q) - ceil(q)); \\ Michel Marcus, Sep 10 2018

a(n) = Sum_{i=1..n} 1 + floor(i/d(i)) - ceiling(i/d(i)), where d(n) is the number of divisors of n.

A281294 Refactorable numbers k such that 2*k + 1 is also a refactorable number.

Original entry on oeis.org

3280, 6160, 8320, 51520, 99904, 174640, 386320, 541840, 883120, 1690960, 2062480, 2365312, 2688880, 2959744, 3077680, 3152560, 3274240, 5375920, 6885760, 8925312, 10030720, 11219584, 11912080, 12058960, 14370160, 15854080, 18966640, 21839440, 22038160, 24787840, 26725360

Offset: 1

Altug Alkan, Jan 19 2017

nonn

If k is in this sequence, then 2*k + 1 must be a square. So this sequence is a subsequence of A046092.

			3280 is a term because 3280 = 2^4 * 5 * 41 is divisible by d(3280) = 2^2 * 5 and 2 * 3280 + 1 = 3^8 is divisible by d(3^8) = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033950, A036896, A046092.

Select[Table[2 n (n + 1), {n, 10^4}], Times @@ Boole@ Thread[Divisible[#, DivisorSigma[0, #]] &@ {#, 2 # + 1}] > 0 &] (* Michael De Vlieger, Jan 19 2017 *)

isA033950(n) = n % numdiv(n)== 0;
is(n) = isA033950(n) && isA033950(2*n+1);

A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).

Original entry on oeis.org

3, 39, 225, 249, 321, 447, 471, 519, 681, 831, 921, 993, 1119, 1191, 1473, 1641, 1671, 1857, 1929, 1983, 2361, 2391, 2463, 2625, 2631, 2913, 3321, 3369, 3561, 3591, 3777, 3807, 3831, 3903, 4119, 4281, 4287, 4359, 4545, 4569, 4791, 5001, 5025, 5079, 5241, 5481

Offset: 1

Jianing Song, Apr 01 2021

Numbers m such that m^2-1 is divisible by d(m^2-1) and m^2 is divisible by d(m^2), d = A000005.
Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. Such k must be of the form m^2-1 for some odd m.
The smallest term not divisible by 3 is a(66) = 9025.
For the first terms we have d(a(n)^2-1) > d(a(n)^2). But this is not always the case. The smallest counterexample is a(30) = 3591, where d(3591^2-1) = 40 and d(3591^2) = 63. The terms m such that d(m^2-1) < d(m^2) are listed in A342970. [Note that d(m^2-1) = d(m^2) is impossible since d(m^2-1) is even and d(m^2) is odd. - Jianing Song, Nov 21 2021]

			39 is a term since 39^2-1 = 1520 is divisible by d(1520) = 20 and 39^2 = 1521 is divisible by d(1521) = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3110 from Jianing Song)
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A033950, A036896, A036898, A114617, A342970.

refQ[n_] := Divisible[n, DivisorSigma[0, n]]; Select[Range[6000], And @@ refQ /@ (#^2 - {1, 0}) &] (* Amiram Eldar, Feb 03 2025 *)

isrefac(n) = ! (n % numdiv(n));
isA342969(n) = (n>1) && isrefac(n^2-1) && isrefac(n^2)

A036898(2*n+1) = A114617(n+1) = a(n)^2 - 1; A036898(2*n+2) = A114617(n+1) + 1 = a(n)^2.

A120319 RF(3): refactorable numbers with smallest prime factor 3.

Original entry on oeis.org

9, 225, 441, 1089, 1521, 2025, 2601, 3249, 4761, 5625, 6561, 7569, 8649, 12321, 15129, 16641, 19881, 25281, 31329, 33489, 35721, 40401, 45369, 47961, 50625, 56169, 62001, 71289, 84681, 91809, 95481, 99225, 103041, 106929, 114921, 145161, 154449, 164025, 168921

Offset: 1

Walter Kehowski, Jun 20 2006

nonn

Numbers that are odd squares, 3 is their smallest prime factor, and are refactorable.

See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 3^(3-1)=9 is the first element. Other elements would also be 3^2*17^2 or 3^16*17^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A016945 and A033950.
Cf. A036896, A036897.
Subsequence of A016946.

with(numtheory); RF3:=[]: p:=3: for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if k mod p = 0 then n:=k^2; t:=tau(n); if (n mod t = 0) then RF3:=[op(RF3),n]; print(ifactor(n)); fi fi; od od;

lista(kmax) = forstep(k = 3, kmax, 6, if(!(k^2 % numdiv(k^2)), print1(k^2, ", "))); \\ Amiram Eldar, Aug 01 2024

a(37)-a(39) from Amiram Eldar, Aug 01 2024

A120320 RF(5): refactorable numbers with smallest prime factor 5.

Original entry on oeis.org

625, 1500625, 9150625, 17850625, 37515625, 52200625, 73530625, 81450625, 174900625, 442050625, 577200625, 1171350625, 1766100625, 1838265625, 2136750625, 3049800625, 4931550625, 7573350625, 8653650625, 12594450625, 15882300625, 17748900625, 21970650625, 24343800625

Offset: 1

Walter Kehowski, Jun 20 2006

nonn

Numbers that are odd squares, 5 is their smallest prime factor, and are refactorable.
See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 5^(5-1) = 625 is the first element. Other elements would also be 5^4*17^4 or 5^16*17^4.
All the terms are of the form 5^2 * A084967(k)^2 = 5^4 * A007310(k)^2. - Amiram Eldar, Aug 01 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A033950 and A084967.

Cf. A007310, A036896, A036897.

with(numtheory); RF5:=[]: p:=5: for w to 1 do for j from 1 to 12^5 do k:=2*j+1; if k mod 3 <> 0 and k mod p = 0 then n:=k^2; t:=tau(n); if (n mod t = 0) then RF5:=[op(RF5),n]; print(ifactor(n)); fi fi; od od;

lista(kmax) = {my(m); for(k = 1, kmax, m = 25*(k\2*6-(-1)^k)^2; if(!(m % numdiv(m)), print1(m, ", ")));} \\ Amiram Eldar, Aug 01 2024

a(37)-a(40) from Amiram Eldar, Aug 01 2024

A120321 RF(7): refactorable numbers with 7 as smallest prime factor.

Original entry on oeis.org

117649, 208422380089, 567869252041, 2839760855281, 5534900853769, 17416274304961, 69980368892329, 104413920565969, 301855146292441, 558845013849409, 743702041351801, 1268163904241521, 2607614922465721

Offset: 1

Walter Kehowski, Jun 21 2006

nonn

Numbers that are odd squares, 7 is their smallest prime factor, and are refactorable.

See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 7^(7-1)=117649 is the first element. Other elements would also be 7^6*17^6 or 7^16*17^6. Here are the prime factorizations for the first 49 elements of RF7: (7^6), (7^6)*(11^6), (7^6)*(13^6), (7^6)*(17^6), (7^6)*(19^6), (7^6)*(23^6), (7^6)*(29^6), (7^6)*(31^6), (7^6)*(37^6), (7^6)*(41^6), (7^6)*(43^6), (7^6)*(47^6), (7^6)*(53^6), (7^6)*(59^6), (7^6)*(61^6), (7^6)*(67^6), (7^6)*(71^6), (7^6)*(73^6), (7^6)*(79^6), (7^6)*(83^6), (7^6)*(89^6), (7^12)*(13^6), (7^6)*(97^6), (7^6)*(101^6), (7^6)*(103^6), (7^6)*(107^6), (7^6)*(109^6), (7^6)*(113^6), (7^6)*(127^6), (7^6)*(131^6), (7^6)*(137^6), (7^6)*(139^6), (7^6)*(11^6)*(13^6), (7^6)*(149^6), (7^6)*(151^6), (7^6)*(157^6), (7^6)*(163^6), (7^6)*(167^6), (7^6)*(13^12), (7^6)*(173^6), (7^6)*(179^6), (7^6)*(181^6), (7^6)*(11^6)*(17^6), (7^6)*(191^6), (7^6)*(193^6), (7^6)*(197^6), (7^6)*(199^6), (7^6)*(11^6)*(19^6), (7^6)*(211^6).

			a(1) = 7^(7-1) = 117649.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A033950 and A084968.

Cf. A036896, A036897.

with(numtheory); p:=7: RF7:=[p^(p-1)]: P:=[seq(ithprime(i),i=2..pi(p)-1)]; for w to 1 do for j from 1 to 12^3 do k:=2*j+1; if andmap(z -> k mod z <> 0, P) then for s from 2 to p-1 by 2 do #accelerate creation n:=7^6*k^s; t:=tau(n); if not n in RF7 and (n mod t = 0) then RF7:=[op(RF7),n]; print(ifactor(n)); fi; od; fi; od od; RF7:=sort(RF7);

cp's OEIS Frontend

A049439 Numbers k such that the number of odd divisors of k is an odd divisor of k.

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A057265 Even refactorable numbers (i.e., the number of divisors is itself a divisor and it is also even).

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Magma

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A181795 Numbers k such that the number of odd divisors of k is an odd divisor of k, and the number of even divisors of k is an even divisor of k.

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A036897 Square root of odd refactorable numbers.

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A208251 Number of refactorable numbers less than or equal to n.

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Maple

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A281294 Refactorable numbers k such that 2*k + 1 is also a refactorable number.

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A342969 Numbers m such that both m^2-1 and m^2 are refactorable numbers (A033950).

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